#8323227

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#45616	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4275 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,4,6)(3,5,7,8)(9,10)(11,19)(12,22)(13,16)(14,20)(15,17)(18,23)(21,24)(25,26,31,34)(27,30,33,39)(28,29,32,38)(35,36,37,40);(3,7)(5,8)(9,11,15,13)(10,16,17,19)(12,20,23,21)(14,22,24,18)(25,27)(26,30)(28,35)(29,36)(31,33)(32,37)(34,39)(38,40);(1,2)(3,5)(4,6)(7,8)(10,17)(14,24)(16,19)(18,22)(27,33)(30,39)(35,37)(36,40);(1,3)(2,5)(4,7)(6,8)(9,12,15,23)(10,18,17,22)(11,21,13,20)(14,16,24,19)(25,28)(26,29)(27,35)(30,36)(31,32)(33,37)(34,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,13,15,11)(10,19,17,16)(12,21,23,20)(14,18,24,22)(25,29)(26,32)(27,36)(28,34)(30,37)(31,38)(33,40)(35,39);(9,14,15,24)(10,20,17,21)(11,22,13,18)(12,16,23,19)(25,30,31,39)(26,33,34,27)(28,36,32,40)(29,37,38,35)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8230423);